You can't hide site requests from within a network such as an office environment as chances are they run directly through the server within the office building so all your HTTP requests got through this server. Chrome incognito mode is simply like private browsing on any other browser it stops the browser saving the search history to the browser. It does not stop your requests being visible to the administration.

Assuming you do this on your own time there is nothing the administration can do. You can not be fired for looking at jobs or because your moving work.

the problem with office computers are that they can contain programs that report your every keystroke,
that said, not everyone employs people to look at it.
if you are below the radar and cover your tracks daily, by running something like ccleaner - from www.piriform.com (assuming there is no corporate spyware) you should be fine.
This cleans your history everytime you run it. (this does not stop an administrator from looking on his proxy server) THIS INFO IS FOR YOUR LOCAL PC ONLY.

Any tripwires set behind the scenes you wont know about. I would suggest you use a personal laptop with a 3g connection and keep it closed when not using it.